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This article is cited in 4 scientific papers (total in 4 papers)
Three and Four-body Systems in One Dimension: Integrability, Superintegrability and Discrete Symmetries
Claudia Chanua, Luca Degiovannib, Giovanni Rastellib a Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via Cozzi 53, 20126 Milano, Italy
b Formerly at Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
Abstract:
Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean space, examples are the Calogero, Wolfes and Tramblay Turbiner Winternitz systems. For some of these systems, we show in a new way how the superintegrability is associated with their dihedral symmetry in the three-dimensional space, the order of the dihedral symmetries being associated with the degree of the polynomial in the momenta first integrals. As a generalization, we introduce the analysis of integrability and superintegrability of four-body systems in one dimension by interpreting them as one-body systems with the symmetries of the Platonic polyhedra in the four-dimensional Euclidean space. The paper is intended as a short review of recent results in the sector, emphasizing the relevance of discrete symmetries for the superintegrability of the systems considered.
Keywords:
superintegrability, higher-degree first integrals, discrete symmetries, Tremblay-Turbiner–Winterniz system.
Received: 11.11.2010 Accepted: 27.02.2011
Citation:
Claudia Chanu, Luca Degiovanni, Giovanni Rastelli, “Three and Four-body Systems in One Dimension: Integrability, Superintegrability and Discrete Symmetries”, Regul. Chaotic Dyn., 16:5 (2011), 496–503
Linking options:
https://www.mathnet.ru/eng/rcd465 https://www.mathnet.ru/eng/rcd/v16/i5/p496
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